I am trying to find the proofs for the following two theorems called the 'projection theorem', which is said to be hold if $x$ and $y$ are jointly normal.
$$E[\tilde{x}\mid \tilde{y} = y] = E[\tilde{x}] + \frac{cov(\tilde{x},\tilde{y})}{var(\tilde{y})}\times(\tilde{y}-E(\tilde{y})),$$ $$var[\tilde{x}\mid \tilde{y}] = var(\tilde{x})-\frac{cov^2(\tilde{x},\tilde{y})}{var(\tilde{y})}.$$
Can anyone provide me the proofs for these two equations? thanks in advance!
Let $a=x-E(x)-\frac {cov(x,y)}{var(y)}(y-E(y))$, $b=y-E(y)$. it is easy to prove a,b are jointly normal and independent. let the joint and marginal pdf of a,b be $g(a,b), g_a(a), g_b(b)$ and the joint and marginal pdf of x, y be $f(x,y), f_x(x), f_y(y)$. Here $g_a$ is the pdf of $N(0,var(x)-\frac{cov(x,y)^2}{var(y)})$
it can be shown that \begin{align} f(x,y)&=g(x-E(x)-\frac {cov(x,y)}{var(y)}(y-E(y)),y-E(y))\\ &=g_a(x-E(x)-\frac {cov(x,y)}{var(y)}(y-E(y)))g_b(y-E(y))\\ &=g_a(x-E(x)-\frac {cov(x,y)}{var(y)}(y-E(y)))f_y(y) \end{align}
So the conditional distribution of x given y has pdf $\frac{f(x,y)}{f_y(y)}=g_a(x-E(x)-\frac {cov(x,y)}{var(y)}(y-E(y)))$, which is the pdf of $N(E(x)+\frac {cov(x,y)}{var(y)}(y-E(y)),var(x)-\frac{cov(x,y)^2}{var(y)})$.